Time changed spherical Brownian motions with longitudinal drifts

Abstract

In this paper, we consider the time change of the diffusion process on the 2-dimensional unit sphere generated by the Laplace-Beltrami operator, perturbed by means of a longitudinal vector field. First, this is done by addressing the problem of finding strong solutions to suitable time-nonlocal Kolmogorov equations, via a spectral decomposition approach. Next, the desired process is constructed as the composition of the aforementioned diffusion process and the inverse of a subordinator, and it is used to provide a stochastic representation of the solution of the involved time-nonlocal Kolmogorov equation, which in turn leads to the spectral decomposition of its probability density function. A family of operators induced by the process is then adopted to provide very weak solutions of the same time-nonlocal Kolmogorov equation with much less regular initial data. From the spectral decomposition results we also get some bounds on the speed of convergence to the stationary state, proving that the process can be considered an anomalous diffusion. These results improve some known ones in terms of both the presence of a perturbation and the lack of regularity of the initial data.